1 Applications of the Calculus The calculus is a mathematical process with many applications. Of interest are those aspects of calculus that enable us to calculate the maximum and minimum value of a function. For example, the maximum value of this function is to be calculated: We know this occurs at

2 x y 200 tangents The slope of tangents to any curve can be calculated using differentiation.

3 At the functions maximum value, the tangent is horizontal and so its slope is 0. This is true for all maximum and minimum values. Using calculus then, involves searching for the places where the slope of the tangent is zero.

4 Rules for Differentiation Notation: The derivative of a function is denoted by

5 Rules for Differentiation

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8 Special Cases: Rules for Differentiation

9 represents the slope of the tangent to the curve y. The introductory example has

10 When the tangent is horizontal (slope = 0) then The maximum occurs at x = 200.

11 Using differentiation to find the maximum and minimum points is a more general method in that it “works” for all curves. Using is a shortcut method that works for quadratic equations (parabolas) only. The equation of the tangent is not difficult to determine. At a point on a curve, the slope is calculated using differentiation and the result is used in the equation

12 Example: Find the equation of the tangent to the curve when x = 2. When x = 2, and the point on the curve is (2,9)

13 So using

14 Example: Find all the points on the curve where the slope is 6.

15 Example: Find the derivative of the following function

16 Example: Find the derivative of the following function

17 Example: Find the derivative of the following function

18 The Derivative as a Measure of Rate of Change The derivative as a measure of the gradient of a tangent to a curve has been discussed. A second interpretation of this process describes the rate of change of some variable.

19 Consider the following physics example: A car’s position (s) is described by the equation, where s is in metres and t is in seconds. After 2 seconds, (t = 2),. The car is 4 metres from some starting point. At t = 10, s = 10 2 = 100 and so on.

20 Looking more closely at the case when t = 10, s = 100, the question is asked “How fast was the car travelling?” One answer is 100m in 10 seconds equals 100/10 = 10 m/s. This describes the average velocity (velocity is a measure of the rate of change of position(s) ). But as with any car journey, the velocity is always changing. Differentiation allows us to calculate the velocity at any time instant.

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22 t (secs)s (m) Average velocity = 100/10 = 10 m/s The derivative describes the rate of change of position (instantaneous velocity) at any time. In summary:

23 Consider the following situation: The total cost function is given by

24 The average cost for producing q items is given by ( is the usual symbol for average) That is,

25 How can the derivative be used and interpreted? describes how quickly cost is changing A comparison shows that cost is changing at a faster rate when q = 10 than when q = 1000.

26 These figures also give an approximate cost to produce the next item. The 11 th item will cost approximately $9990 to produce. The 1001 st item will cost approximately $9000 to produce. When C describes the total cost, then represents the marginal cost. is called the marginal cost function.

27 Marginal cost function: Marginal cost Example:

28 Total cost function Marginal cost function: Example:

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30 There is an equivalent interpretation for the revenue function. If R represents total revenue then is called the marginal revenue. Marginal revenue describes two things: (1). How quickly revenue is changing. (2). Approximate revenue received by selling the next unit.

31 Example: Marginal revenue:

32 The following conclusions can be drawn from these results: (1). Revenue is changing faster at q = 10 than at q = 20. (2). The 11 th item will generate approximately $56 and the 21 st item will earn $52